Question:

From the graph, we have the following signed areas based on the regions bounded by the curve and the x-axis:\n- $\int_a^c f(x) dx = -5$ (area is below the x-axis)\n- $\int_c^d f(x) dx = 20$ (area is above the x-axis)\n- $\int_d^e f(x) dx = -7$ (area is below the x-axis)\n- $\int_e^b f(x) dx = 2$ (area is above the x-axis)\n\n(i) $\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^d f(x) dx + \int_d^e f(x) dx + \int_e^b f(x) dx = -5 + 20 - 7 + 2 = 10$\n\n(ii) $\int_b^a f(x) dx = -\int_a^b f(x) dx = -10$\n\n(iii) $\int_e^c f(x) dx = -\int_c^e f(x) dx = -(\int_c^d f(x) dx + \int_d^e f(x) dx) = -(20 - 7) = -13$\n\n(iv) $\int_a^d f(x) dx + \int_b^e f(x) dx = (\int_a^c f(x) dx + \int_c^d f(x) dx) + (-\int_e^b f(x) dx) = (-5 + 20) + (-2) = 15 - 2 = 13$